A370278 -id:A370278 - cp's OEIS frontend

A370279 Indices of record high values in A370278.

2, 3, 23, 118, 217, 611, 1299, 1949, 3331, 4403

Offset: 1

Zachary DeStefano, Feb 13 2024

			The 118th index of A370278 is the first occurrence of the value 3 in that sequence. All prior values are 0, 1, or 2.

Cf. A370277, A370278.

A370277 Numbers k with the property that Dirichlet's Simultaneous Approximation Theorem applied to Z_k is tight (for d = 3).

Original entry on oeis.org

2, 4, 5, 7, 8, 10, 11, 14, 18, 26, 27, 30, 31, 63, 64, 68, 69, 70, 76, 124, 125, 130, 131, 132, 148, 215, 216, 222, 223, 224, 225, 234, 342, 343, 350, 351, 352, 353

Offset: 1

Zachary DeStefano, Feb 13 2024

Dirichlet's Simultaneous Approximation Theorem applied to Z_k states that for all a_1, a_2, ..., a_d, there exists a nonzero p such that |pa_i| <= k^(1 - 1/d) mod k.
For d = 3, the bound of floor(k^(2/3)) is tight only for specific values of k. That is to say, max_(a_1,a_2,a_3) min_p max_i |pa_i| = floor(k^(2/3)) only for specific values of k. These are those values.
This sequence consists of the indices of the zeros in A370278.
It appears that this sequence contains all integers k such that k or k+1 is a cube.

			For k = 14, floor(k^(2/3)) = 5. Given the triple (1, 3, 5), there is no choice of p such that |p| mod 14, |3p| mod 14, and |5p| mod 14 are all smaller than 5.
p = 1, 3, 5, 9, 11, and 13 results in a simultaneous minimum of 5.

Stack Exchange Dirichlet's Simultaneous Approximation Bound.

Cf. A370278, A370279.

cp's OEIS Frontend

A370279 Indices of record high values in A370278.

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